Fatou's conjecture for quadratic rational maps

Consider complex rational maps of the sphere, in particular quadratic maps. Such a map satisfies axiom A when its non-wandering set is a finite union of basic pieces, and its critical points are the points where its differential fails to be locally invertible. Fatou's conjecture. Structurally stable quadratic maps are those which satisfy axiom A and whose critical points are not periodic. The source explains that the original Fatou conjecture concerned density of axiom A quadratic maps and states that this equivalent formulation has been proved for real quadratic maps, while the supplied text gives no general resolution status.